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79
Probabilistic Principal Component Analysis
 Journal of the Royal Statistical Society, Series B
, 1999
"... Principal component analysis (PCA) is a ubiquitous technique for data analysis and processing, but one which is not based upon a probability model. In this paper we demonstrate how the principal axes of a set of observed data vectors may be determined through maximumlikelihood estimation of paramet ..."
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Cited by 476 (5 self)
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Principal component analysis (PCA) is a ubiquitous technique for data analysis and processing, but one which is not based upon a probability model. In this paper we demonstrate how the principal axes of a set of observed data vectors may be determined through maximumlikelihood estimation of parameters in a latent variable model closely related to factor analysis. We consider the properties of the associated likelihood function, giving an EM algorithm for estimating the principal subspace iteratively, and discuss, with illustrative examples, the advantages conveyed by this probabilistic approach to PCA. Keywords: Principal component analysis
Mixtures of Probabilistic Principal Component Analysers
, 1998
"... Principal component analysis (PCA) is one of the most popular techniques for processing, compressing and visualising data, although its effectiveness is limited by its global linearity. While nonlinear variants of PCA have been proposed, an alternative paradigm is to capture data complexity by a com ..."
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Cited by 398 (6 self)
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Principal component analysis (PCA) is one of the most popular techniques for processing, compressing and visualising data, although its effectiveness is limited by its global linearity. While nonlinear variants of PCA have been proposed, an alternative paradigm is to capture data complexity by a combination of local linear PCA projections. However, conventional PCA does not correspond to a probability density, and so there is no unique way to combine PCA models. Previous attempts to formulate mixture models for PCA have therefore to some extent been ad hoc. In this paper, PCA is formulated within a maximumlikelihood framework, based on a specific form of Gaussian latent variable model. This leads to a welldefined mixture model for probabilistic principal component analysers, whose parameters can be determined using an EM algorithm. We discuss the advantages of this model in the context of clustering, density modelling and local dimensionality reduction, and we demonstrate its applicat...
Principal Components Analysis to Summarize Microarray Experiments: Application to Sporulation Time Series
 in Pacific Symposium on Biocomputing
, 2000
"... A series of microarray experiments produces observations of differential expression for thousands of genes across multiple conditions. It is often not clear whether a set of experiments are measuring fundamentally different gene expression states or are measuring similar states created through diffe ..."
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Cited by 140 (3 self)
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A series of microarray experiments produces observations of differential expression for thousands of genes across multiple conditions. It is often not clear whether a set of experiments are measuring fundamentally different gene expression states or are measuring similar states created through different mechanisms. It is useful, therefore, to define a core set of independent features for the expression states that allow them to be compared directly. Principal components analysis (PCA) is a statistical technique for determining the key variables in a multidimensional data set that explain the differences in the observations, and can be used to simplify the analysis and visualization of multidimensional data sets. We show that application of PCA to expression data (where the experimental conditions are the variables, and the gene expression measurements are the observations) allows us to summarize the ways in which gene responses vary under different conditions. Examination of the components also provides insight into the underlying factors that are measured in the experiments. We applied PCA to the publicly released yeast sporulation data set (Chu et al. 1998). In that work, 7 different measurements of gene expression were made over time. PCA on the timepoints suggests that much of the observed variability in the experiment can be summarized in just 2 components—i.e. 2 variables capture most of the information. These components appear to represent (1) overall induction level and (2) change in induction level over time. We also examined the clusters proposed in the original paper, and show how they are manifested in principal component space. Our results are available on the internet at
Probabilistic nonlinear principal component analysis with Gaussian process latent variable models
 Journal of Machine Learning Research
, 2005
"... Summarising a high dimensional data set with a low dimensional embedding is a standard approach for exploring its structure. In this paper we provide an overview of some existing techniques for discovering such embeddings. We then introduce a novel probabilistic interpretation of principal component ..."
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Cited by 134 (14 self)
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Summarising a high dimensional data set with a low dimensional embedding is a standard approach for exploring its structure. In this paper we provide an overview of some existing techniques for discovering such embeddings. We then introduce a novel probabilistic interpretation of principal component analysis (PCA) that we term dual probabilistic PCA (DPPCA). The DPPCA model has the additional advantage that the linear mappings from the embedded space can easily be nonlinearised through Gaussian processes. We refer to this model as a Gaussian process latent variable model (GPLVM). Through analysis of the GPLVM objective function, we relate the model to popular spectral techniques such as kernel PCA and multidimensional scaling. We then review a practical algorithm for GPLVMs in the context of large data sets and develop it to also handle discrete valued data and missing attributes. We demonstrate the model on a range of realworld and artificially generated data sets.
The benefits of being present: Mindfulness and its role in psychological wellbeing
 Journal of Personality & Social Psychology
, 2003
"... Mindfulness is an attribute of consciousness long believed to promote wellbeing. This research provides a theoretical and empirical examination of the role of mindfulness in psychological wellbeing. The development and psychometric properties of the dispositional Mindful Attention Awareness Scale ..."
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Cited by 83 (3 self)
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Mindfulness is an attribute of consciousness long believed to promote wellbeing. This research provides a theoretical and empirical examination of the role of mindfulness in psychological wellbeing. The development and psychometric properties of the dispositional Mindful Attention Awareness Scale (MAAS) are described. Correlational, quasiexperimental, and laboratory studies then show that the MAAS measures a unique quality of consciousness that is related to a variety of wellbeing constructs, that differentiates mindfulness practitioners from others, and that is associated with enhanced selfawareness. An experiencesampling study shows that both dispositional and state mindfulness predict selfregulated behavior and positive emotional states. Finally, a clinical intervention study with cancer patients demonstrates that increases in mindfulness over time relate to declines in mood disturbance and stress. Many philosophical, spiritual, and psychological traditions emphasize the importance of the quality of consciousness for the maintenance and enhancement of wellbeing (Wilber, 2000). Despite this, it is easy to overlook the importance of consciousness in human wellbeing because almost everyone exercises its primary
Latent variable models
 Learning in Graphical Models
, 1999
"... Abstract. A powerful approach to probabilistic modelling involves supplementing a set of observed variables with additional latent, or hidden, variables. By defining a joint distribution over visible and latent variables, the corresponding distribution of the observed variables is then obtained by m ..."
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Cited by 31 (0 self)
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Abstract. A powerful approach to probabilistic modelling involves supplementing a set of observed variables with additional latent, or hidden, variables. By defining a joint distribution over visible and latent variables, the corresponding distribution of the observed variables is then obtained by marginalization. This allows relatively complex distributions to be expressed in terms of more tractable joint distributions over the expanded variable space. One wellknown example of a hidden variable model is the mixture distribution in which the hidden variable is the discrete component label. In the case of continuous latent variables we obtain models such as factor analysis. The structure of such probabilistic models can be made particularly transparent by giving them a graphical representation, usually in terms of a directed acyclic graph, or Bayesian network. In this chapter we provide an overview of latent variable models for representing continuous variables. We show how a particular form of linear latent variable model can be used to provide a probabilistic formulation of the wellknown technique of principal components analysis (PCA). By extending this technique to mixtures, and hierarchical mixtures, of probabilistic PCA models we are led to a powerful interactive algorithm for data visualization. We also show how the probabilistic PCA approach can be generalized to nonlinear latent variable models leading to the Generative Topographic Mapping algorithm (GTM). Finally, we show how GTM can itself be extended to model temporal data.
Geometric Methods for Feature Extraction and Dimensional Reduction
 In L. Rokach and O. Maimon (Eds.), Data
, 2005
"... Abstract We give a tutorial overview of several geometric methods for feature extraction and dimensional reduction. We divide the methods into projective methods and methods that model the manifold on which the data lies. For projective methods, we review projection pursuit, principal component anal ..."
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Cited by 31 (1 self)
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Abstract We give a tutorial overview of several geometric methods for feature extraction and dimensional reduction. We divide the methods into projective methods and methods that model the manifold on which the data lies. For projective methods, we review projection pursuit, principal component analysis (PCA), kernel PCA, probabilistic PCA, and oriented PCA; and for the manifold methods, we review multidimensional scaling (MDS), landmark MDS, Isomap, locally linear embedding, Laplacian eigenmaps and spectral clustering. The Nyström method, which links several of the algorithms, is also reviewed. The goal is to provide a selfcontained review of the concepts and mathematics underlying these algorithms.
A theory of hyperfinite processes: the complete removal of individual uncertainty via exact LLN
, 1998
"... The aim of this paper is to provide a viable measuretheoretic framework for the study of random phenomena involving a large number of economic entities. The work is based on the fact that processes which are measurable with respect to hyperfinite Loeb product spaces capture the limiting behaviors ..."
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Cited by 20 (10 self)
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The aim of this paper is to provide a viable measuretheoretic framework for the study of random phenomena involving a large number of economic entities. The work is based on the fact that processes which are measurable with respect to hyperfinite Loeb product spaces capture the limiting behaviors of triangular arrays of random variables and thus constitute the `right' class for general stochastic modeling. The primary concern of the paper is to characterize those hyperfinite processes satisfying the exact law of large numbers by using the basic notions of conditional expectation, orthogonality, uncorrelatedness and independence together with some unifying multiplicative properties of random variables. The general structure of the processes is also analyzed via a biorthogonal expansion of the KarhunenLoeve type and via the representation in terms of the simpler hyperfinite Loeb counting spaces. A universality property for atomless Loeb product spaces is formulated to show the abun...
Correlated Bayesian Factor Analysis
, 1998
"... Factor analysis is a method in multivariate statistical analysis that can help scientists determine which variables to study in a field and their relationships. We extend the Bayesian approach to factor analysis developed in 1989 by Press and Shigemasu (henceforth PS89) and revised in 1997 to model ..."
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Cited by 16 (7 self)
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Factor analysis is a method in multivariate statistical analysis that can help scientists determine which variables to study in a field and their relationships. We extend the Bayesian approach to factor analysis developed in 1989 by Press and Shigemasu (henceforth PS89) and revised in 1997 to model correlated observation vectors, factor score vectors, and factor loadings. Further, we place a prior distribution on the number of factors and obtain posterior estimates. Hitherto,